[talanum1 (OP)]

  We start with the real line (a 1-dimensional space). To construct a two dimensional space we identify the zero point, copy the real line and turn the copy through 90 degrees around the point "zero". There is one way to do this.

  Now we scale up all the numbers by one. To construct a three dimensional space we take a copy of the plane and rotate it through 90 degrees around an axis of the plane. There are two ways to do this.

  To construct a four dimensional space we must copy the 3D space and rotate it through 90 degrees around a 2D surface. This is impossible. Nevertheless we can construct a 4D space symbolically by: ℝ×ℝ×ℝ×ℝ, but the analysis shows that this can be done in three ways, since there are three 2D surfaces to rotate around.

We could specify this as: ((ℝ×ℝ)×ℝ)×ℝ or (ℝ×(ℝ×ℝ))×ℝ or (where brackets fails us) ((ℝ)×ℝ×(ℝ))×ℝ.

[evan_au]

I think what you are describing is the mathematical generalization from real numbers → imaginary numbers → quaternions → octonions → sedenions.
- No, they are not alien species (and the last one was new to me)
- They are a way of producing numbers that work in higher dimensions
- And yes, it is possible to go beyond 3 dimensions, with some restrictions

Real numbers (1-dimensional) have two familiar operations: addition (+) and multiplication (*)
- You can swap them around: a + b = b + a; a * b = b * a
- There is a value which leaves the result unchanged: a + 0 = a; a * 1 = a
- And these both have an "inverse" operation: subtraction (-)  and division (/): a + b - b = a; a * b / b = a;
- only "division by zero" is not defined

These rules also apply to imaginary numbers (2 dimensional).

When you try to apply these rules to quaternions (4-dimensional, but usable in 3-dimensional computer graphics), you find that you can't always swap them around: In a computer game, "walking forward, turning left then climbing up" gets you into a different room than "climbing up, turning left and walking forward". So one of the familiar rules from primary school falls by the wayside when you get to 3 dimensions and above.

When you try to apply these rules to octonions (8 dimensional), another rule is inapplicable.

And, apparently, in sedenions, division by zero is defined (which I'm sure must be useful to someone!).

Mathematicians have pet names for which sets of rules are applicable, like "groups", "fields", "rings", "domains", etc;

These names also apply to non-numeric objects like a piece of paper:
- For a rectangular sheet of paper you can define 2 operations: "rotate" or "flip over".
- These operations have an "inverse": The opposite of "rotate clockwise by 90°" is "rotate anticlockwise by 90°"; The opposite of "flip over" is "flip over" (it is its own inverse).
- You can do these two physical operations in different sequences
- And you get a different set of answers (a different mathematics) depending whether the rectangular piece of paper is square (or not), or the same colour on both sides (or not), or whether you only allow rotation by 180°, 90° vs arbitrary angles.
- This whole Field of mathematics is called Group Theory
- Group Theory also applies to subatomic particles...

For more, see the links on page: https://en.wikipedia.org/wiki/Quaternion
https://en.wikipedia.org/wiki/Group_theory